D.K. Gartling is a senior scientist in the Engineering Sciences Center at Sandia Nationa Laboratories, Albuquerque, New Mexico. He earned his B.S. and M.S. in Aerospace Engineering at the University of Texas at Austin and completed the diploma course at th von Karman Institute for Fluid Dynamics in Brussels, Belgium. After completion of his Ph.D. in Aerospace Engineering at the University of Texas at Austin, he joined the technical staff at Sandia National Laboratories. Dr. Gartling was a Visiting Associate Professor in the Mechanical Engineering Department at the University of Sydney, Australia, under a Fulbright Fellowship, and later he was a Supervisor in the Fluid and Thermal Sciences Department at Sandia National Laboratories. Dr. Gartling has published numerous papers dealing with Þnite element model development and finite element analysis of heat transfer and fluid dynamics problems of practical importance. He is the recipient of the 2001 Computational Fluid Dynamics Award from the U.S. Association of Computational Mechanics and is a fellow of the American Society of Mechanical Engineers. Dr. Gartling is presently a member of several professional societies, serves on the editorial boards o several journals, and is the Co-Editor of International Journal for Numerical Methods I Fluids.
The advent of finite element methods has arguably been one of the most important advances in the history of applied sciences, making possible the analysis of thousands of physical phenomena and engineering systems modeled by partial differential equations. The original edition of this book was written as an attempt to put the subject on a solid mathematical foundation, and to demonstrate that the theory of finite elements rests comfortably within the modern theory of PDE's. Since it was published, the field has undergone enormous development and change, and many of the earlier theoretical results have been displaced by deeper and far more general developments. Yet, many of the basic ideas remain unchanged: the notion of weak or generalized solutions, properties of Sobolev spaces, the embedding theories, theorems on existence and uniqueness of solutions, a priori error estimates, affine element families, etc. For this reason, much of the older work is still hoped to have merit beyond its historical value.
During a remarkably short span of years the subject of finite elements has expanded from a collection of effective techniques for solving practical problems in engineering and science to a rich and exciting branch of applied mathematics. The aim of this book is to present the student of engineering science or applied mathematics an introductory account of this mathematical theory.
The book has developed as a result of seminars and courses on finite-element theory taught by the authors at five universities in recent years to students with diverse backgrounds and often modest mathematical preparation. For such an audience, we have found it effective to begin the study with basic mathematical concepts and to systematically build on these the elements of approximation theory, Hilbert spaces, and partial differential equations essential to an understanding of the most important aspects of linear finite-element theory. This book essentially follows this plan. To keep the size and scope of the work within reasonable limits, it has been necessary to omit several important topics in favor of more basic ones. For example, we have not included material on nonlinear problems, integral equations, or eigenvalue problems. However, some of these subjects should be easily mastered by the reader of this book; other subjects must await study in future works.
We owe a great deal to those who developed the mathematical theory of finite elements in recent years. We have been particularly influenced by the work of Ivo Babuška and J. P. Aubin, and we have profited not only in writing this book but also in our own research, from the writings of Philippe Ciarlet, P. A. Raviart, J. L. Lions, and others, and from numerous discussions with our colleague, Ralph Showalter. The first author registers a special note of thanks to the Finite-Element Circus and to certain members of the Circus who have patiently discussed the subject with him; particularly Ivo Babuška, Jim Douglas, Ridgway Scott, Gilbert Strang, Al Schatz, Bruce Kellogg, Mary Wheeler, and James Bramble. We are also thankful for the advice we have received from several colleagues who read an early draft of the manuscript. In particular, we have benefited from the suggestions of John Cannon and Linda Hayes, who read the entire manuscript, and from the comments of Philippe Ciarlet. We also express thanks to M. G. Sheu, N. Kikuchi, and C. T. Reddy who helped with the proofreading. Much of our work on finite-element methods has been supported through the Air Force Office of Scientific Research and the U. S. National Science Foundation. We express our sincere gratitude for this support.
The mathematical theory of finite elements was no exception. By 1965, hundreds of papers on the method had appeared in the engineering literature; it was widely used in industrial applications, was the subject of courses at most technical universities, and was firmly established as a general and powerful method of analysis. However, only a handful of mathematically oriented papers on the method appeared in the engineering literature of the 1960's, and only one or two purely mathematical papers on the subject appeared in 1968 and 1969. A complete mathematical theory began to be pieced together in the 1970's. It was about this time that the strength and elegance of the method and its relation to contemporary research in interpolation theory, splines, and differential equations began to be appreciated by the mathematical world. In a remarkably short time, a virtual flood of mathematical literature appeared on the subject, and a variety of important aspects of the method was quickly put on a sound mathematical footing.
Today the theory has reached a fairly high degree of development, at least as it applies to linear elliptic boundary-value problems, and its foundations are now recognized to be a natural union of spline theory and the modern theory of partial differential equations. In addition, finite-element methods occupy an increasingly important place in modern numerical analysis, and their implementation continues to prompt developments in computational methods and computer software.
As the mathematical theory of finite elements continues to be developed in the framework described above, it becomes more inaccessible to the very practitioners who developed it, not to mention the beginning student who wishes to learn the underlying mathematical features. This book was written with these readers in mind. It represents a systematic introduction to the mathematical theory of finite elements, starting with relatively elementary mathematical concepts and proceeding to an exposé of basic mathematical properties of finite-element approximations of linear boundary-value problems.
This book provides a systematic, highly practical introduction to the use of energy principles, traditional variational methods, and the finite element method for the solution of engineering problems involving bars, beams, torsion, plane elasticity, trusses, and plates. 076b4e4f54